Bound on the projective dimension of three cubics

We show that given any polynomial ring R over a field, and any ideal J in R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question whether ideals generated by three cubic forms can have projective dimension greater than 4, by constructing one with projective dimension equal to 5.Comment: to appear in Journal of Symbolic Computatio

Bound on the multiplicity of almost complete intersections

Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and $(f_1,...,f_N)$ has height $N$, then the multiplicity of $R/I$ is bounded above by $\prod_{i=1}^N d_i - \max\{1, \sum_{i=1}^N (d_i-1) - (d_{N+1}-1) \}$.Comment: 7 pages; to appear in Communications in Algebr

On the projective dimension and the unmixed part of three cubics

Let $R$ be a polynomial ring over a field in an unspecified number of variables. We prove that if $J \subset R$ is an ideal generated by three cubic forms, and the unmixed part of $J$ contains a quadric, then the projective dimension of $R/J$ is at most 4. To this end, we show that if $K \subset R$ is a three-generated ideal of height two and $L \subset R$ an ideal linked to the unmixed part of $K$, then the projective dimension of $R/K$ is bounded above by the projective dimension of $R/L$ plus one.Comment: 23 pages; to appear in Journal of Algebr

Bound on the multiplicity of almost complete intersections

Let R be a polynomial ring over a field of characteristic zero and let I in R be a graded ideal of height N which is minimally generated by N+1 homogeneous polynomials. If I=(f_1,...,f_{N+1}) where f_i has degree d_i and (f_1,...,f_N) has height N, then the multiplicity of R/I is bounded above by product_{i=1}^N d_i - max{ 1, sum_{i=1}^N (d_i-1) - (d_{N+1}-1) }